Probing Biointerfaces: Electrokinetics - Biointerfaces: Where Material Meets Biology - page 59

where the spatial coordinate, x, and the velocity field, v, are normalized
according to X ¼ x/H and V(X) ¼ n(X)/n 0 respectively, with n 0 ¼ DPH 2 /ZL 0 ,
where Z is the dynamic viscosity of the electrolyte. The function f represents
the distribution of polymer segment density across the soft interfacial layer
and it may be derived from polymer theory (e.g., for brushes) or experi-
mentally determined by applying optical and scattering methods. 35-37 As
extensively discussed by Duval and colleagues, 30 the following function
allows for an appropriate modeling of soft interfaces:
f (x) ¼ o
2

d n 3 r 4 n g | 5

x d
a

1 tanh (3 : 2)

In eqn (3.2), the parameter a is the characteristic decay length of the
segment density across the soft layer, and the scalar quantity o ensures that
the total amount of polymer segments within the soft film remains constant
upon structural variations that result from changes in the electrolyte com-
position (pH, salt content) or temperature. 30 In the limit a/d 0, f pertains

to a film with homogeneous segment distribution (step-function like pro-
file). The friction exerted by the polymer segments located at the position
x on the fluid flow is determined by the term (l 0 H) 2 f(X) in the Brinkman
equation. The parameter l 0 stands for the hydrodynamic softness of the film.
Thequantity1/l 0 is the Brinkman length that reflects the characteristic flow
penetration within the soft film in case of homogeneous segment distri-
bution, i.e., a

-

1. The boundaries for the solution of eqn (3.1) are
provided by the no-slip condition at the supporting hard carrier surface (X ¼ 0)
and the symmetry of the flow field with respect to the position X ¼ 1/2. 30

- 0ando -

.

3.2.1.2 Electrostatic Potential
The equilibrium electrical potential distribution, c(x), across the substrate-
soft polymer film-electrolyte solution interface is a function of the local
charge density within the layer. The charge balance has two contributions,
one stemming from the fixed charges within the layer and the other from
mobile electrolyte ions that compensate the charge of the soft layer struc-
ture. These features are taken into account in the non-linear Poisson-
Boltzmann equation that governs the position-dependent potential, c(x),
according to: 32

( )

dX 2 ¼ (kH) 2

X

N

z i c i e z i y(X) þ X

M

d 2 y(X)

r j = F
1 þ 10 e j (pK j pH) e e j y(X)

P i ¼ 1 c i z i f (X) (3 : 3)

i ¼ 1 j ¼ 1

In eqn (3.3), y is the dimensionless potential (y ¼ Fc/RT), k is the re-
ciprocal Debye length, F the Faraday constant, R the gas constant, and T the
temperature. It is assumed that the film carries M types of ionizable groups.
In the limits a 1 the charge source term, r j ¼ 1, ..., M /F,

corresponds to the total concentration of ionisable groups of type j within
the film. The ionisation of the j-th type (j ¼ 1, ..., M) of the functional groups

- 0 and o -